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W2015338532 abstract "We study the effect of the coefficient <svg style=vertical-align:-2.3205pt;width:30.1625px; id=M1 height=15.1125 version=1.1 viewBox=0 0 30.1625 15.1125 width=30.1625 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.162)><path id=x210E d=M499 88q-41 -46 -83.5 -73t-68.5 -27q-41 0 -19 98l52 225q10 43 8 59.5t-16 16.5q-37 0 -112 -65t-117 -135q-14 -42 -38 -179q-36 -7 -75 -20l-7 8l127 594q8 40 2.5 49t-35.5 9h-35l5 25q36 3 71.5 13t56.5 18t25 8q11 0 3 -38l-88 -417h2q77 102 195 167q44 24 79 24
q61 0 30 -130l-52 -220q-8 -33 4 -33q8 0 31 14.5t43 33.5z /></g><g transform=matrix(.017,-0,0,-.017,8.834,12.162)><path id=x28 d=M300 -147l-18 -23q-106 71 -159 185.5t-53 254.5v1q0 139 53 252.5t159 186.5l18 -24q-74 -62 -115.5 -173.5t-41.5 -242.5q0 -130 41.5 -242.5t115.5 -174.5z /></g><g transform=matrix(.017,-0,0,-.017,14.715,12.162)><path id=x1D465 d=M536 404q0 -17 -13.5 -31.5t-26.5 -14.5q-8 0 -15 10q-11 14 -25 14q-22 0 -67 -50q-47 -52 -68 -82l37 -102q31 -88 55 -88t78 59l16 -23q-32 -48 -68.5 -78t-65.5 -30q-19 0 -37.5 20t-29.5 53l-41 116q-72 -106 -114.5 -147.5t-79.5 -41.5q-21 0 -34.5 14t-13.5 37
q0 16 13.5 31.5t28.5 15.5q12 0 17 -11q5 -10 25 -10q22 0 57.5 36t89.5 111l-40 108q-22 58 -36 58q-21 0 -67 -57l-19 20q81 107 125 107q17 0 30 -22t39 -88l22 -55q68 92 108.5 128.5t74.5 36.5q20 0 32.5 -14t12.5 -30z /></g><g transform=matrix(.017,-0,0,-.017,24.218,12.162)><path id=x29 d=M275 270q0 -296 -211 -440l-19 23q75 62 116.5 174t41.5 243t-42 243t-116 173l19 24q211 -144 211 -440z /></g> </svg> of the critical nonlinearity on the number of positive solutions for semilinear elliptic systems. Under suitable assumptions for <svg style=vertical-align:-3.56265pt;width:32.299999px; id=M2 height=16.6625 version=1.1 viewBox=0 0 32.299999 16.6625 width=32.299999 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.162)><path id=x1D453 d=M619 670q0 -13 -9 -26t-18 -19q-13 -10 -25 2q-36 38 -66 38q-31 0 -54.5 -50t-45.5 -185h120l-20 -31l-107 -12q-23 -138 -57 -293q-27 -122 -55 -184.5t-75 -109.5q-60 -61 -114 -61q-25 0 -47.5 15t-22.5 31q0 17 31 44q11 8 20 -1q10 -11 31 -19t35 -8q26 0 47 19
q34 34 71 253l54 315h-90l-3 12l31 30h70q28 138 90 204q35 37 75 57.5t70 20.5q26 0 45 -14t19 -28z /></g><g transform=matrix(.017,-0,0,-.017,10.976,12.162)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,16.857,12.162)><use xlink:href=#x1D465/></g><g transform=matrix(.017,-0,0,-.017,26.36,12.162)><use xlink:href=#x29/></g> </svg>, <svg style=vertical-align:-3.56265pt;width:31.0625px; id=M3 height=16.625 version=1.1 viewBox=0 0 31.0625 16.625 width=31.0625 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.138)><path id=x1D454 d=M546 430q-14 -33 -37.5 -140t-38.5 -209q-29 -195 -156 -289q-29 -21 -70 -37t-78 -16q-62 0 -102.5 37t-40.5 69q0 27 26 48q19 15 25 -3q13 -43 46.5 -69t87.5 -26q55 0 83 21q29 22 50 65t43 140q14 64 29 163h-2q-55 -81 -149 -152q-59 -44 -102 -44q-23 0 -43.5 30
t-20.5 85q0 74 32.5 145.5t84.5 117.5q40 35 100 58.5t117 23.5q21 0 65 -8q30 -6 44 -6zM456 386q-40 16 -81 16q-29 0 -64 -13q-42 -17 -85 -97q-19 -36 -31.5 -86t-12.5 -86q0 -64 30 -64q20 0 52.5 21t60.5 52q74 81 95 123q26 62 36 134z /></g><g transform=matrix(.017,-0,0,-.017,9.735,12.138)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,15.616,12.138)><use xlink:href=#x1D465/></g><g transform=matrix(.017,-0,0,-.017,25.119,12.138)><use xlink:href=#x29/></g> </svg>, and <svg style=vertical-align:-2.3205pt;width:30.1625px; id=M4 height=15.1125 version=1.1 viewBox=0 0 30.1625 15.1125 width=30.1625 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.162)><use xlink:href=#x210E/></g><g transform=matrix(.017,-0,0,-.017,8.834,12.162)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,14.715,12.162)><use xlink:href=#x1D465/></g><g transform=matrix(.017,-0,0,-.017,24.218,12.162)><use xlink:href=#x29/></g> </svg>, we should prove that for sufficiently small <svg style=vertical-align:-3.56265pt;width:53.237499px; id=M5 height=16.6625 version=1.1 viewBox=0 0 53.237499 16.6625 width=53.237499 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.162)><path id=x1D706 d=M529 97q-70 -109 -136 -109q-41 0 -56 94q-23 144 -37 284q-38 -88 -99 -202.5t-93 -156.5q-26 -8 -76 -19l-9 21q71 78 145.5 193t124.5 232q-5 84 -15 128q-12 55 -29.5 75.5t-42.5 20.5q-21 0 -45 -13l-8 24q16 17 46 30t55 13q43 0 70 -46.5t40 -169.5
q27 -249 51 -392q7 -46 23 -46q24 0 70 60z /></g><g transform=matrix(.017,-0,0,-.017,9.446,12.162)><path id=x2C d=M95 130q31 0 61 -30t30 -78q0 -53 -38 -87.5t-93 -51.5l-11 29q77 31 77 85q0 26 -17.5 43t-44.5 24q-4 0 -8.5 6.5t-4.5 17.5q0 18 15 30t34 12z /></g><g transform=matrix(.017,-0,0,-.017,16.143,12.162)><path id=x1D707 d=M538 96q-38 -45 -84 -76.5t-71 -31.5q-44 0 -15 117q12 49 18 89h-2q-99 -128 -163 -178q-36 -28 -70 -28q-26 0 -44 38h-2q-16 -69 -16 -129q0 -43 11 -75.5t24 -42.5l1 -6q-7 -12 -24.5 -23t-37.5 -11q-40 0 -40 76q0 52 35 202l94 407l78 24l5 -5q-32 -122 -72 -310
q-7 -33 0 -55t25 -22q36 0 105.5 75t107.5 145l32 146l75 26h10q-50 -197 -78 -347q-7 -38 6 -38q7 0 31 17t47 39z /></g><g transform=matrix(.017,-0,0,-.017,30.303,12.162)><path id=x3E d=M512 230l-437 -233v58l378 199v2l-378 200v58l437 -233v-51z /></g><g transform=matrix(.017,-0,0,-.017,45.007,12.162)><path id=x30 d=M241 635q53 0 94 -28.5t63.5 -76t33.5 -102.5t11 -116q0 -58 -11 -112.5t-34 -103.5t-63.5 -78.5t-94.5 -29.5t-95 28t-64.5 75t-34.5 102.5t-11 118.5q0 58 11.5 112.5t34.5 103t64.5 78t95.5 29.5zM238 602q-32 0 -55.5 -25t-35.5 -68t-17.5 -91t-5.5 -105
q0 -76 10 -138.5t37 -107.5t69 -45q32 0 55.5 25t35.5 68.5t17.5 91.5t5.5 105t-5.5 105.5t-18 92t-36 68t-56.5 24.5z /></g> </svg>, there are at least <svg style=vertical-align:-0.1638pt;width:34.362499px; id=M6 height=12.4375 version=1.1 viewBox=0 0 34.362499 12.4375 width=34.362499 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,12.162)><path id=x1D458 d=M480 416q0 -21 -18 -41q-9 -11 -17 -7q-20 9 -42 9q-62 0 -140 -78q23 -69 88 -192q17 -31 27 -42t20 -11q16 0 62 46l17 -20q-64 -92 -119 -92q-35 0 -70 66q-41 73 -84 187q-36 -30 -62 -61q-27 -115 -35 -172q-41 -8 -78 -20l-6 6l140 612q7 28 0.5 34t-37.5 7l-34 1
l5 26q38 4 74 13.5t57 17t25 7.5q12 0 4 -32l-104 -443h2q35 38 97 93q39 35 65.5 56t62 41.5t58.5 20.5q19 0 30.5 -10t11.5 -22z /></g><g transform=matrix(.017,-0,0,-.017,12.387,12.162)><path id=x2B d=M535 230h-212v-233h-58v233h-213v50h213v210h58v-210h212v-50z /></g><g transform=matrix(.017,-0,0,-.017,26.139,12.162)><path id=x31 d=M384 0h-275v27q67 5 81.5 18.5t14.5 68.5v385q0 38 -7.5 47.5t-40.5 10.5l-48 2v24q85 15 178 52v-521q0 -55 14.5 -68.5t82.5 -18.5v-27z /></g> </svg> positive solutions of the semilinear elliptic systems <svg style=vertical-align:-3.56265pt;width:334.28751px; id=M7 height=21.1875 version=1.1 viewBox=0 0 334.28751 21.1875 width=334.28751 xmlns:xlink=http://www.w3.org/1999/xlink xmlns=http://www.w3.org/2000/svg> <g transform=matrix(.017,-0,0,-.017,.062,16.7)><path id=x2212 d=M535 230h-483v50h483v-50z /></g><g transform=matrix(.017,-0,0,-.017,10.041,16.7)><path id=x394 d=M600 0h-557v24l268 633l28 8l261 -641v-24zM497 50l-194 489l-196 -489h390z /></g><g transform=matrix(.017,-0,0,-.017,20.937,16.7)><path id=x1D462 d=M515 96q-37 -44 -84.5 -76t-71.5 -32q-12 0 -18 7t-6 34t10 75l19 89h-2q-87 -113 -181 -176q-43 -29 -83 -29q-46 0 -46 63q0 31 9 67l52 222q7 36 -1 36q-10 0 -76 -50l-13 24q47 45 92 71.5t67 26.5q18 0 21 -16.5t-8 -65.5l-56 -242q-16 -67 13 -67q41 0 114 74
t114 146l32 145l74 26h11q-52 -199 -80 -347q-8 -39 6 -39q7 0 32 17.5t47 39.5z /></g><g transform=matrix(.017,-0,0,-.017,34.791,16.7)><path id=x3D d=M535 323h-483v50h483v-50zM535 138h-483v50h483v-50z /></g><g transform=matrix(.017,-0,0,-.017,49.495,16.7)><use xlink:href=#x1D706/></g><g transform=matrix(.017,-0,0,-.017,58.878,16.7)><use xlink:href=#x1D453/></g><g transform=matrix(.017,-0,0,-.017,69.791,16.7)><use xlink:href=#x28/></g><g transform=matrix(.017,-0,0,-.017,75.673,16.7)><use xlink:href=#x1D465/></g><g transform=matrix(.017,-0,0,-.017,85.175,16.7)><use xlink:href=#x29/></g><g transform=matrix(.017,-0,0,-.017,91.057,16.7)><path id=x7C d=M162 -163h-61v866h61v-866z /></g><g transform=matrix(.017,-0,0,-.017,95.527,16.7)><use xlink:href=#x1D462/></g><g transform=matrix(.017,-0,0,-.017,104.673,16.7)><use xlink:href=#x7C/></g> <g transform=matrix(.012,-0,0,-.012,109.15,8.537)><path id=x1D45E d=M474 429q-19 -69 -47 -222l-65 -347q-8 -49 0 -60t49 -16l26 -3l-4 -26l-246 -12l4 25l17 3q37 6 50 20.5t23 60.5l67 321h-2q-65 -79 -150 -138q-69 -47 -104 -47q-28 0 -48.5 32t-20.5 81q0 78 35 148t90 117q67 57 161 77q23 5 58 5q43 0 90 -15zM387 387
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